Clifford's theorem

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This fact is related to: linear representation theory
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Statement

Verbal statement

The restriction of any irreducible complex character of a group, to a normal subgroup, is a multiple of the sum of all conjugates in the whole group of some irreducible character of the normal subgroup.

Statement with symbols, using character-theoretic language

Let $G$ be a finite group and $N$ a normal subgroup of $G$. Let $\chi$ be a complex irreducible character of $G$ and $\mu$ of $N$ such that: $\langle \operatorname{Res}(\chi)_N^G, \mu \rangle \ne 0$

Then: $\operatorname{Res}(\chi)_N^G = a \left(\sum_{i=1}^t \mu^{(g_i)}\right)$

where $g_i \in G$ and $\mu^{(g)}$ denotes the character: $n \mapsto \mu(gng^{-1})$

Further $a$ and $t$ are positive integers dividing the index $[G:N]$. In fact, $t$ is the index of the subgroup $I_G(\mu)$, defined as: $\left \{ g \in G| \mu^{(g)} = \mu \right \}$ $I_G(\mu)$ is termed the inertial subgroup.

Further, $e$ divides the index $[I_G(\mu):N]$.

Statement with symbols, using module-theoretic language

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Particular cases

Conjugacy-closed subgroups

A conjugacy-closed subgroup is a subgroup such that any two elements of the subgroup conjugate in the whole group, are also conjugate in the subgroup. If $N$ is conjugacy-closed, then $I_G(\mu) = G$ for any $\mu$ and thus, in this case, the restriction of the irreducible character from $G$ to $N$ is simply a multiple of $\mu$.